This is for you as well as the newer players...
Turbo Trucker III - 30 Truck Orders In 30 Minutes
As hadaf9 posted, make sure you can deliver all 9 of the initial requests on the board and have plenty of items in your barn. I took it a step further by listing (on paper) all of the items requested by the initial 9 tickets. I wanted to make sure that none of my items were going to be depleted. You may not need to do that with larger capacity barns. When I did it my barn had a small capacity. Also, before you begin, review the achievement and make sure it displays "0/30" for the achievement. That means you're starting with a fresh 30 minute clock versus time is already ticking.
Hyper Harvester III - 200 Fields In 20 Minutes
A key point here is making sure you have enough space in the silo or you're prepared to put crops on your shop as they harvest. It may fill up fast. I used wheat since it harvests the quickest. Do the math before you begin. Make sure you have enough fields on your farm to reach 200, otherwise just wait a bit to level up and acquire more fields.
Hyper Herder III - 60 Animals Products In 30 Minutes
Chickens are your friend in acquiring this. I achieved this before I bought sheep so it can be completed in the earlier levels. I had 10 pigs, 15 cows, and 18 sheep which equals 43 animals. Harvest the chickens again and your done. Make sure you have feed for all the chickens before you begin and make sure you have barn space for the products. If you have more animals, then factor those into the equation and proceed accordingly.
For all of the timed achievements, I recommend starting with a fresh clock. The other achievements are simply a matter of time or money.
Hope this helps.