Here is the full list of all 42 New Yankee: Under the Genie's Thumb achievements.
For fully upgrading the tent
For hiring 11 workers and mercenaries while playing any level
For using the magic cauldron 25 times
For earning a gold trophy on every level
For completing every level
For earning a silver trophy on every level
For building the White Palace
For digging up 25 treasures
For freeing 10 hoopoes
For building a statue of Simurgh
For making pancakes
For drawing mana from wells 50 times
For digging up the ruins of Persepolis
For robbing 25 camels
For chopping down 24 trees on level 31
For using John to defeat 50 cobras
For breaking open 50 barrels
For acquiring 1,500 units of mana while playing level 28
For escorting 10 camels
For acquiring 1,200 units of food while playing level 25
For using Mary to make 25 succubi disappear
For using magic altars 100 times
For capturing Rakhsh
For opening 50 chests
For destroying 10 lotuses
For lowering 10 grates
For freeing 10 workers from cages
For completing the same level three times and improving your results each time
For completing 100 transactions with merchants
For acquiring 1,000 units of gold while playing level 18
For taming Karkadann
For using 10 Sands of Time
For destroying 50 fire spirits
For extinguishing 15 fires
For acquiring 1,500 units of gold while playing any level
For acquiring 1,500 units of wood while playing any level
For using John to destroy 50 skeletons
For acquiring 1,500 units of mana while playing any level
For obtaining the Cup of Jamshid
For acquiring 1,500 units of food while playing any level
For building a house for the raven
For starting a game