The environment is Debian, although the answer will apply to all distributions.
You can also use this command:
dhclient -r interface
dhclient interface
Where interface is the device you want to get a new address for. For instance,
dhclient -r eth0
dhclient eth0
The -r flag forces dhclient to first release any leases you have. The second command requests a new lease.
From man dhclient:
-r Tell dhclient to release the current lease it has from the
server. This is not required by the DHCP protocol, but some
ISPs require their clients to notify the server if they wish
to release an assigned IP address.