On Linux, how can I tell which process is sending Ethernet packets?

I prefer nethogs. It's a small ncurses-based console program that displays per-process network traffic status in a convenient way.

netstat -ptu will give you the owning process ids (along with standard netstat info) for all tcp and udp conections. (Normal users will not be able to id all processes.)

If something is sending out a fair amount of constant traffic you should see it on Recv-Q or Send-Q columns 2 and 3 respectively.

Examples:
Recv-Q
sudo watch -n .1 'netstat -tup | grep -E "^[tc,ud]p[6]{0,1}" | sort -nr -k2'

Send-Q
sudo watch -n .1 'netstat -tup | grep -E "^[tc,ud]p[6]{0,1}" | sort -nr -k3'

If you suspect that that process is being triggered by another process ps axf.