If you're using Bash, you can run disown -h job

disown

disown [-ar] [-h] [jobspec ...]

Without options, each jobspec is removed from the table of active jobs. If the -h option is given, the job is not removed from the table, but is marked so that SIGHUP is not sent to the job if the shell receives a SIGHUP. If jobspec is not present, and neither the -a nor -r option is supplied, the current job is used. If no jobspec is supplied, the -a option means to remove or mark all jobs; the -r option without a jobspec argument restricts operation to running jobs.

For Git Bash

If you are running msysgit (I am assuming you are) and are looking to run Git Bash (I recommend it over TortoiseGit, but I lean to the CLI more than GUI now), you need to figure out what your home directory is for Git Bash by starting it then type pwd (On Windows 7, it will be something like C:\Users\phsr I think). While you're in Git Bash, you should mkdir .ssh.

After you have the home directory, and a .ssh folder under that, you want to open PuTTYgen and open the key (.ppk file) you have previously created. Once your key is open, you want to select Conversions -> Export OpenSSH key and save it to HOME\.ssh\id_rsa. After you have the key at that location, Git Bash will recognize the key and use it.

Note: Comments indicate that this doesn't work in all cases. You may need to copy the OpenSSH key to Program Files\Git\.ssh\id_rsa (or Program Files (x86)\Git\.ssh\id_rsa).

For TortoiseGit

When using TortoiseGit, you need to set the SSH key via pacey's directions. You need to do that for every repository you are using TortoiseGit with.