Process Monitor equivalent for Linux?

The grandaddy of all process monitors is top, and many system monitoring tools are called top. For example, there's iotop to watch disk I/O, atop for a bunch of system resources, powertop for power consumption.

If you want more detailed information, it's not tracked by default. To watch what a particular process is doing, call strace on it. For example, if you're only interested in filesystem accesses:

strace -s9999 -efile command_name    # trace a program during its whole execution
strace -s9999 -efile -p1234          # trace a running program with the given PID

strace is specific to Linux, but other systems have a similar tool: truss on Solaris, ktrace or dtrace under *BSD, etc.

To watch what's happening to a particular file or in a particular directory or directory tree, use the inotify facility.

inotifywait -m .

Again, the facility is specific to Linux, but most other unices have a similar system, e.g. kqueue under *BSD, and FAM (originally from SGI but now available as an API on many systems).

To watch all the system calls under Linux, you can use the audit subsystem. It's relatively recent and there's not much literature on the topic; search for auditctl or read the auditctl man page. There are a couple of examples on this site: tracking file accesses, tracking process execution.

The console standby for this is top, but there are alternatives like my favorite htop that give you a little more display flexibility and allow you a few more operations on the processes.

A less interactive view that is better for use in scripts would be the ps program and all it's relatives.

Edit: Based on your clarified question, you might note that strace handles watching system calls made by a given process including all read-write operations and os function calls. You can activate it on the command line before the program you want to track or attach to a running process by hitting s on a process selected in htop.

You may want to take a look at lsof and strace. Sysinternal's Process Monitor is actually Filemon and Regmon combined with a some additional enhancements. The top command is more like Process Explorer.