A process for pyrolyzing a hydrocarbon feedstock containing undesirable coke precursors in an olefins pyrolysis furnace comprising: a) feeding the feedstock to a first stage preheater provided in a convection zone of the furnace, and heating said feedstock within the first stage preheater to produce a heated gas-liquid mixture, b) withdrawing the heated gas-liquid mixture from the first stage preheater to a first vapor-liquid separator, c) separating and removing at least a portion of the gas from the liquid in the first vapor-liquid separator, heating the gas in a vapor phase preheater provided in said convection zone, feeding a first portion of the heated gas into a radiant zone of the pyrolysis furnace and pyrolyzing the gas to produce olefins, and feeding a second portion of the heated gas to a second vapor-liquid separator, d) withdrawing the liquid from the first vapor-liquid separator, and feeding the removed liquid to a thermal-cracking zone, and subjecting the removed liquid to mild thermal-cracking conditions to produce a thermally-cracked product having a significantly diminished heavy fraction containing components with normal boiling points above 537 °C., e) withdrawing the thermally-cracked product from the thermal-cracking zone to the second vapor-liquid separator, f) vaporizing and removing at least a portion of the thermally-cracked components with normal boiling points below 537 °C from the liquid fraction in the second vapor-liquid separator, feeding the removed gas into a radiant zone of the pyrolysis furnace, and pyrolyzing the gas to produce olefins, and g) removing the remaining liquid fraction from the second vapor-liquid separator, wherein said mild thermal-cracking conditions include temperatures in the range of about 425°C to about 525°C, a pressure in the range of from about 2.8 Bar to about 22.1 Bar, and wherein the liquid in said thermal-cracking zone is maintained at the thermal-cracking conditions within the thermal-cracking zone for a time period of from about 10 seconds to about 960 seconds, and wherein the pyrolytic cracking conditions include a pyrolytic cracking temperature of from about 700°C to about 900°C, a pyrolytic cracking pressure of from about 15 psia to about 30 psia, and wherein the gaseous fractions are exposed to the pyrolytic cracking conditions within the radiant zone for a pyrolytic cracking time period upwardly to about 10 seconds. a) feeding the feedstock to a first stage preheater provided in a convection zone of the furnace, and heating said feedstock within the first stage preheater to produce a heated gas-liquid mixture, b) withdrawing the heated gas-liquid mixture from the first stage preheater to a first vapor-liquid separator, c) separating and removing at least a portion of the gas from the liquid in the first vapor-liquid separator, heating the gas in a vapor phase preheater provided in said convection zone, feeding a first portion of the heated gas into a radiant zone of the pyrolysis furnace and pyrolyzing the gas to produce olefins, and feeding a second portion of the heated gas to a second vapor-liquid separator, d) withdrawing the liquid from the first vapor-liquid separator, and feeding the removed liquid to a thermal-cracking zone, and subjecting the removed liquid to mild thermal-cracking conditions to produce a thermally-cracked product having a significantly diminished heavy fraction containing components with normal boiling points above 537 °C., e) withdrawing the thermally-cracked product from the thermal-cracking zone to the second vapor-liquid separator, f) vaporizing and removing at least a portion of the thermally-cracked components with normal boiling points below 537 °C from the liquid fraction in the second vapor-liquid separator, feeding the removed gas into a radiant zone of the pyrolysis furnace, and pyrolyzing the gas to produce olefins, and g) removing the remaining liquid fraction from the second vapor-liquid separator,