What TFTP client should I use to flash my device?

Which ever you want! Some suggestions are given below:

GNU/Linux

Some people suggest atftp with GNU/Linux
Windows

DD-WRT tftp GUI
Server is the IP address of your bootloader tftp server
Password is typically blank
Select the firmware file
Set retries to 20 or more (most of the time you get it in 3)
Click on Upgrade and it will constantly retry until it gets it
Windows 2000 and Windows XP have a built-in TFTP client and it can be used to flash with OpenWrt firmware.
tftpd32 (use client mode!) from http://perso.wanadoo.fr/philippe.jounin/tftpd32.html
TFTP command line client short Instructions

Open a command window (cmd.exe)
"tftp -i <bootloader IP tftp server address> PUT OpenWrt-gs-code.bin"
Now you may plug in the router (unplug it first if it was plugged).
Note that some bootloaders do not respond to ping.