I know this is an old post but it is still very relevant. I have found that modern browsers support rfc5987, which allows utf-8 encoding, percentage encoded (url-encoded). Then Naïve file.txt becomes:

Content-Disposition: attachment; filename*=UTF-8''Na%C3%AFve%20file.txt

Safari (5) does not support this. Instead you should use the Safari standard of writing the file name directly in your utf-8 encoded header:

Content-Disposition: attachment; filename=Naïve file.txt

IE8 and older don't support it either and you need to use the IE standard of utf-8 encoding, percentage encoded:

Content-Disposition: attachment; filename=Na%C3%AFve%20file.txt

In ASP.Net I use the following code:

string contentDisposition;
if (Request.Browser.Browser == "IE" && (Request.Browser.Version == "7.0" || Request.Browser.Version == "8.0"))
    contentDisposition = "attachment; filename=" + Uri.EscapeDataString(fileName);
else if (Request.Browser.Browser == "Safari")
    contentDisposition = "attachment; filename=" + fileName;
else
    contentDisposition = "attachment; filename*=UTF-8''" + Uri.EscapeDataString(fileName);
Response.AddHeader("Content-Disposition", contentDisposition);

I tested the above using IE7, IE8, IE9, Chrome 13, Opera 11, FF5, Safari 5.

Update November 2013:

Here is the code I currently use. I still have to support IE8, so I cannot get rid of the first part. It turns out that browsers on Android use the built in Android download manager and it cannot reliably parse file names in the standard way.

string contentDisposition;
if (Request.Browser.Browser == "IE" && (Request.Browser.Version == "7.0" || Request.Browser.Version == "8.0"))
    contentDisposition = "attachment; filename=" + Uri.EscapeDataString(fileName);
else if (Request.UserAgent != null && Request.UserAgent.ToLowerInvariant().Contains("android")) // android built-in download manager (all browsers on android)
    contentDisposition = "attachment; filename=\"" + MakeAndroidSafeFileName(fileName) + "\"";
else
    contentDisposition = "attachment; filename=\"" + fileName + "\"; filename*=UTF-8''" + Uri.EscapeDataString(fileName);
Response.AddHeader("Content-Disposition", contentDisposition);

The above now tested in IE7-11, Chrome 32, Opera 12, FF25, Safari 6, using this filename for download: 你好abcABCæøåÆØÅäöüïëêîâéíáóúýñ½§!#¤%&()=`@£$€{[]}+´¨^~'-_,;.txt

On IE7 it works for some characters but not all. But who cares about IE7 nowadays?

This is the function I use to generate safe file names for Android. Note that I don't know which characters are supported on Android but that I have tested that these work for sure:

private static readonly Dictionary<char, char> AndroidAllowedChars = "abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ._-+,@£$€!½§~'=()[]{}0123456789".ToDictionary(c => c);
private string MakeAndroidSafeFileName(string fileName)
{
    char[] newFileName = fileName.ToCharArray();
    for (int i = 0; i < newFileName.Length; i++)
    {
        if (!AndroidAllowedChars.ContainsKey(newFileName[i]))
            newFileName[i] = '_';
    }
    return new string(newFileName);
}

@TomZ: I tested in IE7 and IE8 and it turned out that I did not need to escape apostrophe ('). Do you have an example where it fails?

@Dave Van den Eynde: Combining the two file names on one line as according to RFC6266 works except for Android and IE7+8 and I have updated the code to reflect this. Thank you for the suggestion.

@Thilo: No idea about GoodReader or any other non-browser. You might have some luck using the Android approach.

@Alex Zhukovskiy: I don't know why but as discussed on Connect it doesn't seem to work terribly well.

man curl:

   -H/--header <header>
          (HTTP)  Extra header to use when getting a web page. You may specify
          any number of extra headers. Note that if you should  add  a  custom
          header that has the same name as one of the internal ones curl would
          use, your externally set header will be used instead of the internal
          one.  This  allows  you  to make even trickier stuff than curl would
          normally do. You should not replace internally set  headers  without
          knowing  perfectly well what you're doing. Remove an internal header
          by giving a replacement without content on the  right  side  of  the
          colon, as in: -H "Host:".

          curl  will  make sure that each header you add/replace get sent with
          the proper end of line marker, you should thus not  add  that  as  a
          part  of the header content: do not add newlines or carriage returns
          they will only mess things up for you.

          See also the -A/--user-agent and -e/--referer options.

          This option can be used multiple times to add/replace/remove  multi-
          ple headers.

Example:

curl --header "X-MyHeader: 123" www.google.com

You can see the request that curl sent by adding the -v option.