Here are the best of breed – my favorite cross-platform applications.
A nice, full featured graphical text editor with syntax highlighting and many other bells and whistles is Vim, or Vi IMproved – a programmers text editor. I mainly use SciTe which is based on Scintilla. For source code compilers, I use MinGW, WinAVR (avr-gcc), and Yagarto (GNU-ARM). I sometimes use Code::Blocks, the cross platform cross-platform IDE built around wxWidgets.
If you like to play or edit sound, Audacity is a good, simple audio editor suitable for recording, mixing, and editing ordinary tracks of CD-quality audio. VLC is a free cross-platform media player that supports a large number of multimedia formats without the need for additional codecs.
Games I enjoy playing are Tux Racer, BZFlag, Chromium BSU, Nethack, Never Ball, and Critical Mass. There are lots of games, such as Tux Paint, written using the Simple DirectMedia Layer, an open source, cross-platform multimedia library designed to provide fast access to the graphics framebuffer and audio device.
An open source, cross-platform native UI framework is wxWidgets (formerly called wxWindows).
There are some applications that are useful on Win32 where an equivalent application is already included in any major Linux distribution. I use FileZilla for FTP client and FTP server, wheras on Kubuntu Linux I use Konqueror for an FTP client and vsftpd for an FTP server. I use PuTTY for an SSH client on Windows, and Konsole on Kubuntu. I use WinSCP for a secure shell file manager under Windows, and Konqueror under Kubuntu. I use TortoiseSVN and TortoiseCVS and TortoiseHG for source code control, and mostly command line svn or cvs or hg or git clients under Ubuntu. PDF Creator is a tool for creating PDF documents just by printing them from any Windows application. To compare or “diff” files, I use WinMerge on Windows, Kompare on Kubuntu, or Meld on Ubuntu. I use 7-zip for archiving on Windows.