Equivalent of SysInternals FileMon for Linux?
You're looking for strace. Have a look here: https://wiki.ubuntu.com/Strace
It depends on what you want:
In the large, you want to look at inotify to see all file accesses that any process makes.
In the small, strace will let you watch the syscalls a particular process makes. Strace is pretty awesome. You can trace a process's calls to 'open' by doing
strace -f -eopen $cmd, for instance. The man page has full details on syntax, of course.
strace in front of an starting application is good to watch what the app is doing.
lsof is nice to see which files an already running app is using.
lsof -ni:22 shows which process is using Port 22.
SGI has a tool that you might want to try: http://oss.sgi.com/projects/fam/