Q:

Story With Character Named Briar

I am trying to remember a young adult series I read perhaps 15 years ago, but I don't remember many details.  One of the characters was named Briar.  As I recall, there are four main characters that channel elemental power, and they have a meditation routine to unlock this power where they inhale and exhale to a seven count.  Sorry it's not much to go on, I will try to add more details if I remember any. 

A:

You're looking for the Circle of Magic books by Tamora Pierce. 
There are two quartets (as Pierce is prone to writing books in sets of four) and a final stand-alone book to wrap up the series. 
The first quartet starts with the books:

Sandry's Book
Tris's Book
Daja's Book
Briar's Book

The second quartet, The Circle Opens continues their stories as they venture out into the world. 

A:

This sounds like Sandry's Book by Tamora Pierce

Niklaren (Niko) Goldeye finds Lady Sandrilene fa Toren, Trisana
  Chandler, Daja Kisubo, and Briar Moss, four young misfits from four
  different classes, and brings them to Winding Circle Temple in Emelan. 
  They find themselves housed together as they did not "fit in" when
  they slept in the dormitories for everyone in Discipline Cottage to
  learn and use their new-found magical abilities.  All four have ambient
  magic, as opposed to academic magic, and the power they use comes from
  ordinary things all around them.  Sandry has magic with threads, Tris
  with weather, Daja with smithing, and Briar with plants.