Is seismology really the only way to measure the thickness of Earth's layers? Answers from these two questions: 
How can we guess the size of the earths inner core? 
&amp; 
How can we measure the thickness of the earths mantle? 
use seismic waves as the method for detirmining size and thickness of sections in the earth.

Is this the only way of measuring the inner sections of the earth?

Bonus Points: Where did the idea of using seismic waves to measure earth's layers start?
Robert Mallet is sometimes referred to as the "father of seismology." He was helped (around 1850) by the newly created science of photography, which enabled him to create seismic "maps" of the earth, at different depths.

He has started life as an iron engineer/manufacturer, but turned his talents to seismology after a number of devastating earthquakes in Europe. His earlier work with iron gave him a background in the use of explosives to "open up" the earth at critical points.

Paleoseismology is a related field that relies on paleontology to form estimates of various properties of the earth.