You're using Putty. All you have to do is start your remote miners with the nohup command added to them. Example:
nohup ./bfgminer -o stratum+tcp://stratum.somepool.com:3333 -u workerName -p workerPassword &
If you were to use screen as was suggested earlier, you'd have to install it on the remote machine. Not sure what Linux flavor those remote machines are running, but if they're on a Debian-derived distro, you'd just do this:
sudo apt-get install screen
Then, you'd start your miners like this:
sudo screen -dmS MyMiners ./bfgminer -o stratum+tcp://stratum.somepool.com:3333 -u workerName -p workerPassword
This starts up a session in the background. To access it, you'd just do this:
sudo screen -dR MyMiners
Then, once you're in the screen, to get out of it, all you need to do is hold down the control key and hit a then d.
And with this method, can I just set it and forget it?
Setup, install, run and never login to it again?
Pretty much.
Also, something to consider, as I mentioned earlier, my uncle has provided me access to its server infrastructure in Brazil, but he has given me this access under the form of 1500 DIFFERENT VPS Logins, which means that the 7 Mh/s that I mentioned earlier is only possible by being connected to 1500 INDIVIDUAL SSH sessions.
I had to MANUALLY install cpuminer in 1500 different sessions, which in of itself took me nearly 4 days.
Please tell me I'm not reading this correctly or that you've misrepresented your hash rate, or you're mining something other than BTC. 1500 sessions. 7MH/s??? I get far greater rate than that out of my MacBookPro running CPU miner.