Cannot play random Cataclysm dungeon

Solution 1:

To queue for a random dungeon with the LFG tool, you will have to meet the minimum item level requirements for all Cataclysm dungeons in that particular mode. If you queue for a specific dungeon, you simply need to meet the average item level requirement for that dungeon.

The 'average item level' (ilvl) is a computed number that represents the quality of all of your gear.

According to Wowpedia:

Normal Dungeons

• Blackrock Caverns: 226
• Throne of the Tides: 226
• The Stonecore: 272
• The Vortex Pinnacle: 272
• Grim Batol: 305
• Halls of Origination: 305
• Lost City of the Tol'vir: 305

Heroic Dungeons

All pre-zandalari heroics require a minimum item level of 329.

Heroic Rise of Zandalari Dungeons

These heroics require a minimum item level of 346.

Solution 2:

Change the type from "Random ..." to "Specific Dungeons", then checkout the Cataclysm section, hover the listed dungeons to see why you can play them (they should be locked).

Solution 3:

Unless they patched it, in addition to minimum equipment levels, you also need to physically discover the location of the entrance of the instance in order to queue for it.