How can I connect three monitors to my MacBook Pro?

With one minor exception, the same rules we learned for the M1 and M2 still apply to the M3 models. Display interfaces are generated deep inside the system on a chip. How many there are depends on the exact type of Processor:

M3 (plain) like the 13-in and 15-in supports up to ONE External fully hardware-accelerated external display.

Except the M3 MacBook Air models with 13 or 15-in display, (and certain M3 plain MacBook Pro models when running 14.6 or later) which can support a second external display instead of the built-in display when you close the cover on the built-in display.

M3 PRO processor supports up to Two fully hardware-accelerated external displays.

M3 MAX processor supports up to Four fully hardware-accelerated external displays.

M3 ULTRA processor supports up to Eight fully hardware-accelerated external displays

MacBook Pro (14-inch, 2021) - Technical Specifications - Apple Support

MacBook Pro (16-inch, 2021) - Technical Specifications - Apple Support

"Up to two external displays with up to 6K resolution at 60Hz at over a billion colors (M1 Pro)"

MacBook Pro (14-inch, M3 Pro or M3 Max, Nov 2023) - Technical Specifications - Apple Support

MacBook Pro (16-inch, Nov 2023) - Technical Specifications - Apple Support

"M3 Pro

Up to two external displays with up to 6K resolution at 60Hz over Thunderbolt, or one external display with up to 6K resolution at 60Hz over Thunderbolt and one external display with up to 4K resolution at 144Hz over HDMI

One external display supported at 8K resolution at 60Hz or one external display at 4K resolution at 240Hz over HDMI"

exactly what model Macs are these, and what processor on each {plain, PRO, MAX, Ultra}

is each display on a separate cable?