In the US, the Bishops have said that a bow of head is the required sign of reverence to be made before receiving Holy Communion, so no, walking up in procession is not “reverent enough”.
As for the universal norm, I posted a bit of it in my reply to benedictgal, but I’ll show the whole thing here; this is the Latin (universal) GIRM 160:
Sacerdos deinde accipit patenam vel pyxidem, et accedit ad communicandos, qui de more processionaliter appropinquant. Non licet ipsis fidelibus panem consecratum neque calicem sacrum per semetipsos accipere eo minus de manu in manum inter se transmittere. Fideles communicant genuflexi vel stantes, prout Conferentia Episcoporum statuerit. Cum autem stantes communicant, commendatur ut debitam reverentiam, ab iisdem normis statuendam, ante susceptionem Sacramenti faciant.
Roughly translated: “The priest then holds the paten or ciborium, and approaches the communicants, who as a rule approach in a procession. It is not licit for the faithful to take the consecrated bread nor the sacred chalice by themselves let alone pass them to one another by hand. The faithful communicate (i.e. receive communion) kneeling or standing, as established by the Conference of Bishops. However when communicating standing, it is recommended that the appropriate reverence, according to the same norms to be established, be made before receiving the Sacrament.”
This is how it reads (as amended in red) in the US Missal:The priest then takes the paten or ciborium and goes to the communicants, who, as a rule, approach in a procession.
The faithful are not permitted to take the consecrated bread or the sacred chalice by themselves and, still less, to hand them from one to another. The norm for reception of Holy Communion in the dioceses of the United States is standing. Communicants should not be denied Holy Communion because they kneel. Rather, such instances should be addressed pastorally, by providing the faithful with proper catechesis on the reasons for this norm.
When receiving Holy Communion, the communicant bows his or her head before the Sacrament as a gesture of reverence and receives the Body of the Lord from the minister. The consecrated host may be received either on the tongue or in the hand, at the discretion of each communicant. When Holy Communion is received under both kinds, the sign of reverence is also made before receiving the Precious Blood.