There is no syntax in C++ nor C# for the second method you mentioned.

There's nothing wrong with your first method. If however you have very big ranges, just use a series of if statements.

This syntax is from the Visual Basic Select...Case Statement:

Dim number As Integer = 8
Select Case number
    Case 1 To 5
        Debug.WriteLine("Between 1 and 5, inclusive")
        ' The following is the only Case clause that evaluates to True.
    Case 6, 7, 8
        Debug.WriteLine("Between 6 and 8, inclusive")
    Case Is < 1
        Debug.WriteLine("Equal to 9 or 10")
    Case Else
        Debug.WriteLine("Not between 1 and 10, inclusive")
End Select

You cannot use this syntax in C#. Instead, you must use the syntax from your first example.

You can leave out the newline which gives you:

case 1: case 2: case 3:
   break;

but I consider that bad style.

With C#9 came the Relational Pattern Matching. This allows us to do:

switch (value)
{
    case 1 or 2 or 3:
      // Do stuff
      break;
    case 4 or 5 or 6:
      // Do stuff
      break;
    default:
        // Do stuff
        break;
}

In deep tutorial of Relational Patter in C#9

Pattern-matching changes for C# 9.0

Relational patterns permit the programmer to express that an input value must satisfy a relational constraint when compared to a constant value

.NET Framework 3.5 has got ranges:

Enumerable.Range from MSDN

you can use it with "contains" and the IF statement, since like someone said the SWITCH statement uses the "==" operator.

Here an example:

int c = 2;
if(Enumerable.Range(0,10).Contains(c))
    DoThing();
else if(Enumerable.Range(11,20).Contains(c))
    DoAnotherThing();

But I think we can have more fun: since you won't need the return values and this action doesn't take parameters, you can easily use actions!

public static void MySwitchWithEnumerable(int switchcase, int startNumber, int endNumber, Action action)
{
    if(Enumerable.Range(startNumber, endNumber).Contains(switchcase))
        action();
}

The old example with this new method:

MySwitchWithEnumerable(c, 0, 10, DoThing);
MySwitchWithEnumerable(c, 10, 20, DoAnotherThing);

Since you are passing actions, not values, you should omit the parenthesis, it's very important. If you need function with arguments, just change the type of Action to Action<ParameterType>. If you need return values, use Func<ParameterType, ReturnType>.

In C# 3.0 there is no easy Partial Application to encapsulate the fact the the case parameter is the same, but you create a little helper method (a bit verbose, tho).

public static void MySwitchWithEnumerable(int startNumber, int endNumber, Action action){ 
    MySwitchWithEnumerable(3, startNumber, endNumber, action); 
}

Here an example of how new functional imported statement are IMHO more powerful and elegant than the old imperative one.

Here is the complete C# 7 solution...

switch (value)
{
   case var s when new[] { 1,2,3 }.Contains(s):
      // Do something
      break;
   case var s when new[] { 4,5,6 }.Contains(s):
      // Do something
      break;
   default:
      // Do the default
      break;
}

It works with strings too...

switch (mystring)
{
   case var s when new[] { "Alpha","Beta","Gamma" }.Contains(s):
      // Do something
      break;
...
}

Another option would be to use a routine. If cases 1-3 all execute the same logic then wrap that logic in a routine and call it for each case. I know this doesn't actually get rid of the case statements, but it does implement good style and keep maintenance to a minimum.....

[Edit] Added alternate implementation to match original question...[/Edit]

switch (x)
{
   case 1:
      DoSomething();
      break;
   case 2:
      DoSomething();
      break;
   case 3:
      DoSomething();
      break;
   ...
}

private void DoSomething()
{
   ...
}

Alt

switch (x)
{
   case 1:
   case 2:
   case 3:
      DoSomething();
      break;
   ...
}

private void DoSomething()
{
   ...
}

In C# 7 we now have Pattern Matching so you can do something like:

switch (age)
{
  case 50:
    ageBlock = "the big five-oh";
    break;
  case var testAge when (new List<int>()
      { 80, 81, 82, 83, 84, 85, 86, 87, 88, 89 }).Contains(testAge):
    ageBlock = "octogenarian";
    break;
  case var testAge when ((testAge >= 90) & (testAge <= 99)):
    ageBlock = "nonagenarian";
    break;
  case var testAge when (testAge >= 100):
    ageBlock = "centenarian";
    break;
  default:
    ageBlock = "just old";
    break;
}

One lesser known facet of switch in C# is that it relies on the operator= and since it can be overriden you could have something like this:

string s = foo();

switch (s) {
  case "abc": /*...*/ break;
  case "def": /*...*/ break;
}

gcc implements an extension to the C language to support sequential ranges:

switch (value)
{
   case 1...3:
      //Do Something
      break;
   case 4...6:
      //Do Something
      break;
   default:
      //Do the Default
      break;
}

Edit: Just noticed the C# tag on the question, so presumably a gcc answer doesn't help.

In C# 8.0 you can use the new switch expression syntax which is ideal for your case.

var someOutput = value switch
{
    >= 1 and <= 3 => <Do some stuff>,
    >= 4 and <= 6 => <Do some different stuff>,
    _ => <Default stuff>
};

I think this one is better in C# 7 or above.

switch (value)
{
    case var s when new[] { 1,2 }.Contains(s):
    // Do something
     break;

    default:
    // Do the default
    break;
 }

You can also check Range in C# switch case: Switch case: can I use a range instead of a one number Or if you want to understand basics of C# switch case

You can also have conditions that are completely different

            bool isTrue = true;

            switch (isTrue)
            {
                case bool ifTrue when (ex.Message.Contains("not found")):
                case bool ifTrue when (thing.number = 123):
                case bool ifTrue when (thing.othernumber != 456):
                    response.respCode = 5010;
                    break;
                case bool ifTrue when (otherthing.text = "something else"):
                    response.respCode = 5020;
                    break;
                default:
                    response.respCode = 5000;
                    break;
            }

A more beautiful way to handle that

if ([4, 5, 6, 7].indexOf(value) > -1)
    //Do something

You can do that for multiple values with the same result

Just to add to the conversation, using .NET 4.6.2 I was also able to do the following. I tested the code and it did work for me.

You can also do multiple "OR" statements, like below:

            switch (value)
            {
                case string a when a.Contains("text1"):
                    // Do Something
                    break;
                case string b when b.Contains("text3") || b.Contains("text4") || b.Contains("text5"):
                    // Do Something else
                    break;
                default:
                    // Or do this by default
                    break;
            }

You can also check if it matches a value in an array:

            string[] statuses = { "text3", "text4", "text5"};

            switch (value)
            {
                case string a when a.Contains("text1"):
                    // Do Something
                    break;
                case string b when statuses.Contains(value):                        
                    // Do Something else
                    break;
                default:
                    // Or do this by default
                    break;
            }

We can also use this approach to achieve Multiple cases in switch statement... You can use as many conditions as you want using this approach..

    int i = 209;
    int a = 0;
    switch (a = (i>=1 && i<=100) ? 1 : a){    
    case 1:
        System.out.println ("The Number is Between 1 to 100 ==> " + i);
        break;
    default:
        switch (a = (i>100 && i<=200) ? 2 : a) {
            case 2:
                System.out.println("This Number is Between 101 to 200 ==> " + i);
                break;
        
            default:
                switch (a = (i>200 && i<=300) ? 3 : a) {
                    case 3:
                        System.out.println("This Number is Between 201 to 300 ==> " + i);
                        break;
                
                    default:
                        // You can make as many conditions as you want;
                        break;
                }
        }
        
    }