A method of operating a burner comprising the steps of: (a) delivering an air or other oxygen-containing gas through a flow passageway surrounded by a burner wall, said burner wall having a forward end which is in communication with an interior of a fired heating system in which combustion takes place, said burner wall having a longitudinal axis which extends through said flow passageway, and said flow passageway having a longitudinally forward discharge opening at said forward end of said burner wall from which said air or other oxygen-containing gas is discharged into a combustion zone, (b) delivering a burner fuel to a fuel discharge ring which is located inside of said interior of said fired heating system, wherein said fuel discharge ring entirely surrounds, laterally, said flow passageway, said fuel discharge ring is positioned outside of said flow passageway, and said fuel discharge ring is positioned longitudinally rearward of said forward end of said burner wall, said fuel discharge ring having (i) a plurality of fuel discharge openings formed through a forward surface of said fuel discharge ring, which surround said flow passageway and are oriented in a direct or angled forward direction to deliver said burner fuel outside of said burner wall to said combustion zone or (ii) a single fuel discharge slot formed through said forward surface of said fuel discharge ring, which surrounds said flow passageway and is oriented in a direct or angled forward direction to deliver said burner fuel outside of said burner wall to said combustion zone, and (c) ejecting from each of said plurality of said fuel discharge openings formed through said forward surface of said fuel discharge ring, or from said single fuel discharge slot formed through said forward surface of said fuel discharge ring, a free jet ejection stream of said burner fuel which travels, prior to combustion, along a free jet flow path outside of said burner wall and through a flue gas in said interior of said fired heating system, wherein the free jet flow path begins at said fuel discharge ring and extends from said fuel discharge ring to said combustion zone which begins at or proximate to said forward end of said burner wall in said interior of said fired heating system. (a) delivering an air or other oxygen-containing gas through a flow passageway surrounded by a burner wall, said burner wall having a forward end which is in communication with an interior of a fired heating system in which combustion takes place, said burner wall having a longitudinal axis which extends through said flow passageway, and said flow passageway having a longitudinally forward discharge opening at said forward end of said burner wall from which said air or other oxygen-containing gas is discharged into a combustion zone, (b) delivering a burner fuel to a fuel discharge ring which is located inside of said interior of said fired heating system, wherein said fuel discharge ring entirely surrounds, laterally, said flow passageway, said fuel discharge ring is positioned outside of said flow passageway, and said fuel discharge ring is positioned longitudinally rearward of said forward end of said burner wall, said fuel discharge ring having (i) a plurality of fuel discharge openings formed through a forward surface of said fuel discharge ring, which surround said flow passageway and are oriented in a direct or angled forward direction to deliver said burner fuel outside of said burner wall to said combustion zone or (ii) a single fuel discharge slot formed through said forward surface of said fuel discharge ring, which surrounds said flow passageway and is oriented in a direct or angled forward direction to deliver said burner fuel outside of said burner wall to said combustion zone, and (c) ejecting from each of said plurality of said fuel discharge openings formed through said forward surface of said fuel discharge ring, or from said single fuel discharge slot formed through said forward surface of said fuel discharge ring, a free jet ejection stream of said burner fuel which travels, prior to combustion, along a free jet flow path outside of said burner wall and through a flue gas in said interior of said fired heating system, wherein the free jet flow path begins at said fuel discharge ring and extends from said fuel discharge ring to said combustion zone which begins at or proximate to said forward end of said burner wall in said interior of said fired heating system.