Q:

What would be Linux equivalent of GetProcessIoCounters? 

Here's a link to Windows documentation. 
Basically I would like to get similar data, but on Linux.  If not all is possible, then at least some parts. 

A:

If you enable CONFIG\_TASK\_IO\_ACCOUNTING, you will have the information available in /proc/<pid>/io.  This is available since kernel 2.6.20, but not normally enabled by default (However, in Ubuntu 8.04 it seems like it is enabled). 
You can read about the various data items in Documentation/filesystems/proc.txt in the kernel source tree.  Especially section 2.14 should be of interest.