Apple Cinema will not display an image on Mac Pro

Can you find a model number on the back? Apple made a number of displays that look like that, but they vary, sometimes widely, in their capabilities and connections. But if it has a port that looks like Thunderbolt, most likely it's mini-DisplayPort:

https://en.wikipedia.org/wiki/Mini_DisplayPort

in which case that's probably an A1267 LED Cinema Display (24-inch). Specs here:

LED Cinema Display - Technical Specifications - Apple Support

MacTracker indicates that the following displays had DVI, two USB 2.0 ports, and two FireWire 400 ports.

• Cinema Display (20-inch DVI)
• Cinema Display (23-inch DVI)
• Cinema Display (30-inch DVI)
• Cinema Display (20-inch DVI Late 2005)
• Cinema Display (23-inch DVI Late 2005)
• Cinema Display (30-inch DVI Late 2005)

The 20-inch displays had 1680x1050 pixels, the 23-inch ones had 1920x1200 pixels, and the 30-inch ones had 2560x1600 pixels. For the 30-inch ones, you would need a dual-link DVI connection.

A model number of A1081 makes it one of the two 20-inch ones.

How Many and what kind of cable-ends are provided in the multiple-connector lead-in cord?

There are TWO displays, commonly mistaken for each other that appear to have a Thunderbolt-2 interface.

The Apple Thunderbolt Display 27-in A1407 has a lead in cord with ONLY TWO connectors -- Thunderbolt-2 and MagSafe. This display works great with the Apple ThunderBolt-3 <-> Thunderbolt-2 adapter, US$50.

In addition to the built-in Thunderbolt cable, the display also has a built-in MagSafe-2 connector that charges an older Thunderbolt-equipped MacBook Air or MacBook Pro.

from:

https://everymac.com/monitors/apple/thunderbolt/specs/apple-thunderbolt-display-27-inch-specs.html

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Cinema display LED 27-in A1316 from 2011 features lead-in cord with three connectors -- Mini DisplayPort, MagSafe-2 and a USB-A pigtail. It does NOT work with the Thunderbolt adapter, but works fine with a USB-C to miniDisplayPort adapter.

It includes three USB 2.0 ports and a convenient single cable with THREE connectors (Mini DisplayPort, MagSafe-2, and USB 2.0), a built-in iSight camera with a mic and a 2.1 speaker system.

from:

https://everymac.com/monitors/apple/studio_cinema/specs/apple-led-cinema-display-27-inch-specs.html

OK, that's an 20" Apple Cinema Display from 2004, capable of a resolution of 1680 × 1050. The video input is DVI-D  and it has two FireWire 400 ports and 2 USB 2.0 ports. Both of the standard configurations of the 2009 Mac Pro came with graphics cards that have one mini-DisplayPort and one DVI, so you'll need either an appropriate DVI cable, or a mini-DisplayPort to DVI adapter and then also a DVI cable.

If you have either of those and you still can't get a picture, it's possible that either the graphics card or the display has failed.

Hope this helps.

4K would work just as well as HD.

You would need a different adapter, but HDMI <-> DVI is a "just wires" (cheap) adapter if you want to try that.

the downside is that the more complex the experiment, the less we can learn , especially if it does not work.

is the display seen by the Mac across the interface?

 menu > about this Mac > (System Report) > Graphics&Displays

...

as I said above, the one with three:

"works fine with a USB-C to miniDisplayPort adapter"

tell readers more about your display.

translucent Ice and snow surround, aluminum surround?

approximate diagonal screen size?

any model numbers you can find?

DVI uses the same Higher Voltage levels as HDMI, so it is more failure-prone, It is possible that port burned out on the graphics card, but a Mini Displayport adapter to DVI would work fine.