Today are property lines still ‘shot’ with a transept and chain or GPS in heavy woodland?

In today's world usually a combination of a robotic total station and GPS.
GPS is typically not reliable in a very heavily wooded situation, especially
with a thick overhead canopy. This will vary by season obviously.
A transit and chain is not used if ever anymore. I have used a staff
compass and chain a time or two to try and retrace very old surveys.
Hats off to the old surveyors who did the original surveys with this
equipment, they were better men than me.

Zim

Like Zim said, GPS and Robotic Total Stations are the norm... however when dealing with heavy brush and a thick canopy of tree cover the conventional total station and a 2 man (or sometimes 3 man) crew is necessary. We still run lines for timber companies out here, and the conventional total station is really the only option for that type of work.

If you have a specific question regarding your property lines, please contact a local licensed land surveyor.

Zim gave you a great answer. My suggestion is to hire a surveyor with a good reputation in the community and trust him to use the right tool for the job.