Nope, generally that's not true, TSWeb works with IE, FirefoX web browsers (tested by me) :) but to be able to run any RemoteApp you need to have Remote Desktop Client which is compatible with RDC 6.1 in Windows and as I know that version
is ONLY in Windows :]
So, you cannot run RemoteApps on Linux servers, because of missing RDC not IE :)
@update: And of course, if you wish to use Remote Desktop within TS/RD Web Access, you need IE to support ActiveX
@update2: My mistake ;) TSWeb works itself but do not display any shared apps :) So, IE and RDC are required to work with RemoteApps :) Sorry for inconvenience.