First, I set up the styling (no scrollbar, font, font size, background, and foreground colors):
xterm +sb -fa monaco -fs 10 -bg black -fg white
Next, I redirected the output and backgrounded the process:
xterm +sb -fa monaco -fs 10 -bg black -fg white > /dev/null 2>&1 &
This worked well for quite a while, but when I spawn a shell in an arbitrary directory, I wanted my shell to start in home so I added:
eval $( cd ; xterm +sb -fa monaco -fs 10 -bg black -fg white > /dev/null 2>&1 & )
Finally, I wanted to fully disown the new xterm from the shell I spawned it from. Therefore, my .bash_aliases file now has:
alias term='eval $( cd ; xterm +sb -fa monaco -fs 10 -bg black -fg white > /dev/null 2>&1 & disown %1 )'
Now I can cleanly spawn a new terminal that sends no output to the existing shell.