The government of Quintana Roo state, home to beach resorts Cancun and Tulum, said no one was hurt during the passage of Grace, which struck the Mexican coast as a Category 1 Hurricane in the morning before weakening to a tropical storm inland.
Heading westward, Grace was expected to hit the coast of Veracruz State as a hurricane late on Friday, the Miami-based National Hurricane Center (NHC) said.
There were warnings of hurricane conditions and dangerous storm surge.The NHC said Grace would dump 4-8 inches (10-20 cm) of rain over the Yucatan Peninsula through Friday, and up to 12 inches in some areas.
The heavy rainfall would likely cause areas of flash and urban flooding, it added.Mexican officials said preparations had been made for the hurricane's arrival, with dozens of military and rescue workers as well as staff from the national power utility, the Comision Federal Electricidad, gearing up to help.
Grace unleashed downpours and flooding over Haiti and Jamaica earlier this week.
By Thursday afternoon it was about 80 miles east-northeast of Campeche, with top sustained winds of 60 mph (97 kph), moving west at 18 mph (29 kph), the NHC said.