I also believe James is right here. They do not need to close simultaneously. However, there is a limited time frame you have for closing on the new properties. Best check with an accountant just to be sure.
All 1031 exchanges must follow the same Internal Revenue guidelines. For example, once the seller's property is sold, the seller must identify their replacement property or properties within 45 days (following the identification rules that the IRS has set out as to number and types of properties). The final purchase of the replacement property or properties must take place within 180 days from the date of the sale of the initial property.