In Acts 10:35, what did Peter mean when he stated that
those who work righteousness is accepted of God?

     Peter declared in Acts 10:34-35, “Of a truth I perceive that
God is no respecter of persons: (35) But in every nation he that
feareth him, and worketh righteousness, is accepted with him.”
 
We learn several important truths from these scriptures:
     First, God is no respecter of persons.  He doesn’t have
different plans of salvation.  God’s plan of salvation is one
plan and equally applies to all accountable people (Mark 16:
15-16).
     Second, the term “feareth” means “respect” or
“reverence.”  We must reverence God if we are to be accepted
with Him.
     Third, what does the expression “worketh righteousness”
mean?  First of all, it is clear that we must “work.”  There is
something we must do (cf. Acts 2:40; Phil. 2:12).  Second of
all, Peter made it clear that we must work “righteousness.”  
Working righteousness is applicable to every nation, that is
every one.  Working righteousness is very simply working God’
s righteous plan to save man.  This is made very clear in the
book of Romans.  According to Rom. 1:16-17, “the
righteousness of God” is revealed in the Gospel.  This is not
referring to an attribute of God, but rather to God’s righteous
plan that he has revealed in the Gospel, which is designed, if
worked (Acts 10:35) makes men righteous.  According to Rom.
3:21, “the righteousness of God . . . is manifested.”  Rom. 3:
22 states,
“Even the righteousness of God which is by faith of
Jesus Christ unto all and upon all them that believe: for there is
no difference.”
 This verse is parallel with Acts 10:35.  The
righteousness of God is “by faith.”  Acts 10:35 teaches that
one must “work” righteousness.  Hence it is not faith
“alone.”  Rather, it is a faith that works (Gal. 5:6; Heb. 11).  
Rom. 10:3 brings the matter into focus:
“For they (the Jews)
being ignorant of God’s righteousness, and going about to
establish their own righteousness, have not submitted themselves
unto the righteousness of God.”
 Certainly, we can’t be saved
“by works of righteousness which we have done” (Titus 3:5).  
However, we are saved by working God’s righteous plan.