What (if anything) is the Ubuntu equivalent of network-scripts?

Ubuntu puts all the interfaces into /etc/network/interfaces , the syntax is similar to RHEL for most of what you want.

With Debian/Ubuntu you can run commands from that file with pre-up, post-up, pre-down, post-down.

For your bridge

sudo apt-get -y install bridge-utils uml-utilities

sample /etc/network/interfaces

auto lo
iface lo inet loopback

auto br0
iface br0 inet static
address 192.168.0.10
netmask 255.255.255.0
broadcast 192.168.0.255
gateway 192.168.0.1
bridge_ports eth0
bridge_stp off
bridge_maxwait 5

If you have a more specific question, or get stuck, post your RHEL config and we can help you convert it.

See man interfaces