While doing the switch from passwords to public keys I also found many newer SSH clients options now exist in Windows. Here are the main ones:
Git for Windows, which you may already have installed, has ssh and its related features available within the Git Bash tool. If you're using a passphrase for your key then you'll also find it useful to set up the ssh-agent to auto-launch.
If you install the Windows Subsystem for Linux then from a command prompt you can run:
wsl ssh myserver
Note that I use an SSH config file to simplify the connection command, so I don't need to enter the full hostname or username.
However, you probably wouldn't want to install the whole of WSL just for a little SSH client, especially as Windows 10 now has an SSH client built in. See their OpenSSH blog article for details.
After setting that up, if necessary (it's installed by default in recent releases of Windows 10), you can run the following from a command prompt:
ssh myserver
This last option has, at present, some minor bugs, such as the inability to resize the window after an SSH session has commenced. This bug is described in TTY PTY support in Windows OpenSSH.